On the Neumann (p, q)-eigenvalue problem in Hölder singular domains

In the article we study the Neumann ( p , q )-eigenvalue problems in bounded Hölder γ γ -singular domains Ω γ⊂ Rⁿ Ω γ ⊂ R n . In the case 1<p<∞ 1 < p < ∞ and 1<q<p*γ 1 < q < p γ ∗ we prove solvability of this eigenvalue problem and existence of the minimizer of the associated variational problem. In addition, we establish some regularity results of the eigenfunctions and some estimates of ( p , q )-eigenvalues.